Step 1: The Dirty Bits
1. One Arduino "Whatever" (Uno, Mega, Leonardo, Lovitz, etc).
2. One small bread board (included in above kit)
3. One BLE Mini Bluetooth module from redbearlab.com
4. One Philips 12 volt 3 watt LED porcelain bulb
5. One small servo
6. One 12 volt power supply (at least 1 amp)
6. One lamp with shade
7. Assorted: aluminum sheet (soda can), wire, cardboard, tape, blade/scissors, zip ties
8. Lastly, you'll need an iPhone and an Apple developer account, or some other method of communicating with the Bluetooth module.
Red Bear Lab has an example app on the app store called "BLE Arduino". You can control the servo from here, but you'll likely want to modify and build your own version of the code which pulls in weather for your area. If this is too much to take on for some, I totally understand. You can just as well turn this project into a clever shadow clock or other indicator (insert imagination here).
I'd release the app I've modified onto the app store, but honestly, I'm afraid it would get poor reviews. ;)
Step 2: Down to the Wires
From my count, you'll need 9 wires. Mouse over each in the picture for a description of where they are going to end up.